9.1 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Convection | 6/7 | https://en.wikipedia.org/wiki/Convection | reference | science, encyclopedia | 2026-05-05T10:54:52.394392+00:00 | kb-cron |
Δ
ρ
=
ρ
0
β
Δ
T
{\displaystyle \Delta \rho =\rho _{0}\beta \Delta T}
where
ρ
0
{\displaystyle \rho _{0}}
is the reference density, typically picked to be the average density of the medium,
β
{\displaystyle \beta }
is the coefficient of thermal expansion, and
Δ
T
{\displaystyle \Delta T}
is the temperature difference across the medium. The general diffusivity,
D
{\displaystyle D}
, is redefined as a thermal diffusivity,
α
{\displaystyle \alpha }
.
D
=
α
{\displaystyle D=\alpha }
Inserting these substitutions produces a Rayleigh number that can be used to predict thermal convection.
Ra
=
ρ
0
g
β
Δ
T
L
3
α
μ
{\displaystyle {\textbf {Ra}}={\frac {\rho _{0}g\beta \Delta TL^{3}}{\alpha \mu }}}
=== Turbulence === The tendency of a particular naturally convective system towards turbulence relies on the Grashof number (Gr).
G
r
=
g
β
Δ
T
L
3
ν
2
{\displaystyle Gr={\frac {g\beta \Delta TL^{3}}{\nu ^{2}}}}
In very sticky, viscous fluids (large ν), fluid motion is restricted, and natural convection will be non-turbulent. Following the treatment of the previous subsection, the typical fluid velocity is of the order of
g
Δ
ρ
L
2
/
μ
{\displaystyle g\Delta \rho L^{2}/\mu }
, up to a numerical factor depending on the geometry of the system. Therefore, Grashof number can be thought of as Reynolds number with the velocity of natural convection replacing the velocity in Reynolds number's formula. However In practice, when referring to the Reynolds number, it is understood that one is considering forced convection, and the velocity is taken as the velocity dictated by external constraints (see below).
=== Behavior === The Grashof number can be formulated for natural convection occurring due to a concentration gradient, sometimes termed thermo-solutal convection. In this case, a concentration of hot fluid diffuses into a cold fluid, in much the same way that ink poured into a container of water diffuses to dye the entire space. Then:
G
r
=
g
β
Δ
C
L
3
ν
2
{\displaystyle Gr={\frac {g\beta \Delta CL^{3}}{\nu ^{2}}}}
Natural convection is highly dependent on the geometry of the hot surface, various correlations exist in order to determine the heat transfer coefficient. A general correlation that applies for a variety of geometries is
N
u
=
[
N
u
0
1
2
+
R
a
1
6
(
f
4
(
P
r
)
300
)
1
6
]
2
{\displaystyle Nu=\left[Nu_{0}^{\frac {1}{2}}+Ra^{\frac {1}{6}}\left({\frac {f_{4}\left(Pr\right)}{300}}\right)^{\frac {1}{6}}\right]^{2}}
The value of f4(Pr) is calculated using the following formula
f
4
(
P
r
)
=
[
1
+
(
0.5
P
r
)
9
16
]
−
16
9
{\displaystyle f_{4}(Pr)=\left[1+\left({\frac {0.5}{Pr}}\right)^{\frac {9}{16}}\right]^{\frac {-16}{9}}}
Nu is the Nusselt number and the values of Nu0 and the characteristic length used to calculate Re are listed below (see also Discussion):
Warning: The values indicated for the Horizontal cylinder are wrong; see discussion.
== Natural convection from a vertical plate == One example of natural convection is heat transfer from an isothermal vertical plate immersed in a fluid, causing the fluid to move parallel to the plate. This will occur in any system wherein the density of the moving fluid varies with position. These phenomena will only be of significance when the moving fluid is minimally affected by forced convection. When considering the flow of fluid is a result of heating, the following correlations can be used, assuming the fluid is an ideal diatomic, has adjacent to a vertical plate at constant temperature and the flow of the fluid is completely laminar. Num = 0.478(Gr0.25) Mean Nusselt number = Num = hmL/k where
hm = mean coefficient applicable between the lower edge of the plate and any point in a distance L (W/m2. K) L = height of the vertical surface (m) k = thermal conductivity (W/m. K) Grashof number = Gr =
[
g
L
3
(
t
s
−
t
∞
)
]
/
v
2
T
{\displaystyle [gL^{3}(t_{s}-t_{\infty })]/v^{2}T}
where
g = gravitational acceleration (m/s2) L = distance above the lower edge (m) ts = temperature of the wall (K) t∞ = fluid temperature outside the thermal boundary layer (K) v = kinematic viscosity of the fluid (m2/s) T = absolute temperature (K) When the flow is turbulent different correlations involving the Rayleigh Number (a function of both the Grashof number and the Prandtl number) must be used. Note that the above equation differs from the usual expression for Grashof number because the value
β
{\displaystyle \beta }
has been replaced by its approximation
1
/
T
{\displaystyle 1/T}
, which applies for ideal gases only (a reasonable approximation for air at ambient pressure).
== Pattern formation ==